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Shape Convergence in Conformal Tiling

Banff International Research Station for Mathematical Innovation and Discovery

(Joint work with Phil Bowers, Florida State) The famous "Penrose" tiling is perhaps the most well known hierarchical, aperiodic tiling of the plane. We consider this and other infinite tilings generated by subdivision rules. However, we put conformal structure rather than euclidean structure on the tiles, giving so-called "conformal tilings". Conformal tiling is determined by combinatorics alone, and is not limited by the rigid geometric constraints of classical tilings, thus it brings up new issues in tiling theory. This talk will rely heavily on tiling images. Among other things, those images suggest that aggregates of conformal tiles may converge in shape to their classical euclidean counterparts. This raises an interesting issue: how can rigid euclidean shapes be encoded in abstract combinatorics? This is a new side of tiling theory with many open questions.

Mathematics; Dynamical systems and ergodic theory; Ordinary differential equations; Dynamical systems

video/mp4

Author affiliation: University of Tennessee

2017-01-27

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/59966

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/